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This major new textbook for undergraduates in mathematics and physics is a thorough, self-contained and highly readable account of a subject many students find difficult. The author's clear and systematic style promotes a good understanding of the subject: each concept is motivated and illustrated by worked examples, while problem sets provide plenty of practice for understanding and technique. Computer-based problems, some suitable for projects, are also included. The book is structured to make learning the subject easy; there is a natural progression from core topics to more advanced ones, and harder topics are treated with particular care. A theme of the book is the importance of conservation principles. These appear first in vectorial mechanics where they are proved and applied to problem solving. They reappear in analytical mechanics, where they are shown to be related to symmetries of the Lagrangian, culminating in Noether's theorem.

Suitable for a wide range of undergraduate courses in mathematics and physics departments; assumes no prior knowledge of the subject

Profusely illustrated and thoroughly class-tested, with a clear direct style that makes the subject easy to understand: all concepts are motivated and illustrated by the many worked examples included

Model solutions for teachers available from our online resources page

Douglas Gregory is a Professor of Mathematics at the University of Manchester. He is a researcher of international standing in the field of elasticity and has held visiting positions at New York University, The University of British Columbia and the University of Washington. He is a highly regarded teacher of applied mathematics: this his first book, is the product of many years of teaching experience.

Contents
Prefac e ....................................... xi
1 Newtonian mechanics of a single particl e 1
1 The algebr a and calculus of vectors 3
1. 1 Vector s an d vecto r quantities . . . .. . . . . . . . . . . . . . . . . . . . 3
1. 2 Linear operations: a + b an d λ a .................... . 5
1. 3 The scala r produc t a · b .......................... 10
1. 4 The vecto r produc t a × b ........................ . 13
1. 5 Tripl e products . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 15
1.6 Vec tor functi ons of a sca lar variable ................... . 16
1.7 Tangent and normal vec tors to a curve ................... 18
Problem s .................................. 22
2 Velocity, acceleratio n and scalar angula r velocity 25
2.1 S t raight li ne moti on of a parti cle ...................... 25
2.2 General moti on of a parti cle ....................... . 28
2.3 Parti cle moti on in polar co-ordinates ................... 32
2. 4 Rigi d body rotatin g abou t a fixed axis . . . . . . . . . . . . . . . . . . . 36
2. 5 Rigi d body in planar motion . . . .. . . . . . . . . . . . . . . . . . . . 38
2.6 Reference fr ames in relat ive moti on ................... . 40
Problem s .................................. 43
3 Newton’s law s of motion and th e la w of gr avitatio n 50
3.1 Newton’s laws of moti on .......................... 50
3.2 Inerti al fr ames and the law of inerti a ................... . 52
3. 3 The law of mutual interaction; mass an d forc e . . . . . . . . . . . . . . . 54
3. 4 The law of multipl e interactions . .. . . . . . . . . . . . . . . . . . . . 57
3.5 Cent re of mass ............................... 58
vi Contents
3.6 The law of gravit ati on ........................... 59
3.7 Gravit ati on by a di st ributi on of mass ................... 60
3. 8 The principle of equivalence an d g ................... . 67
Problem s .................................. 71
4 Problem s in particl e dynamic s 73
4. 1 Rectilinear motion in a forc e field . . . . . . . . . . . . . . . . . .. . . 74
4. 2 Constrained rectilinea r motion . . . . . . . . . . . . . . . . . . . .. . . 78
4. 3 Motion throug h a resistin g medium . . . . . . . . . . . . . . . . .. . . 82
4. 4 Projectile s . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .. . . 88
4.5 Ci rcular moti on ............................... 92
Problem s .................................. 98
5 Linear oscillations 105
5.1 Body on a spring .............................. 105
5.2 Cl assica l simple harmonic moti on .................... . 107
5.3 Damped simple harmonic moti on .................... . 109
5.4 Driven (f orce d) moti on ........................... 112
5.5 A simple sei smograph ........................... 120
5. 6 Couple d oscillations an d norma l modes . . . . . . . . . . . . . . .. . . 121
Problem s .................................. 126
6 Energy conservatio n 131
6.1 The energy principle ........................... . 131
6. 2 Energy conser vatio n in rectilinea r motion . . . . . . . . . . . . . .. . . 133
6. 3 Genera l feature s of rectilinea r motion . . . . . . . . . . . . . . . .. . . 136
6. 4 Energy conser vatio n in a conser vat ive field ............... . 140
6. 5 Energy conser vatio n in constrained motion . . . . . . . . . . . . .. . . 145
Problem s .................................. 151
7 Orbits in a centra l field 155
7. 1 The one-body proble m – Newton’s equations . . . . . . . . . . . .. . . 157
7.2 General nature of orbit al moti on ...................... 159
7.3 The path equati on ............................ . 164
7.4 Nea rly ci rcular orbit s ........................... . 167
7. 5 The attract ive inverse square field .................... . 170
7. 6 Spac e tr avel – Hohmann transfe r orbit s .................. 177
7. 7 The repuls ive inverse square field .................... . 179
7.8 Rutherf ord sca tt ering ........................... . 179
Appendix A The geometry of conics ...................... 184
Appendix B The Hohmann orbi t is optima l . . . . . . . . . . . . . . .. . . 186
Problem s .................................. 188
Contents vii
8 Non-linear oscillations and phase spac e 194
8. 1 Periodi c non-linea r oscillations . .. . . . . . . . . . . . . . . . . . . . 194
8.2 The phase plane ( ( x
1
, x
2
)–plane ) .................... . 199
8. 3 The phas e plan e in dynamics (( x , v )–plane) ............... . 202
8. 4 Poinca r ´e-Bendixso n theorem: limi t cycle s . . . . . . . . . . . . . . . . . 205
8. 5 Driven non-linea r oscillations . . .. . . . . . . . . . . . . . . . . . . . 211
Problem s .................................. 214
2 Multi-particle system s 219
9 The energy principle 221
9. 1 Configurations an d degree s of freedo m . . . . . . . . . . . . . . . . . . 221
9. 2 The energy principle fo r a syste m .................... . 223
9. 3 Energy conser vatio n fo r a syste m .................... . 225
9. 4 Kinetic energy of a rigi d body . . .. . . . . . . . . . . . . . . . . . . . 233
Problem s .................................. 241
10 The linear momentum principle 245
10.1 Linea r momentum ............................ . 245
10.2 The li nea r moment um pr i nci pl e ...................... 246
10.3 Moti on of t he ce nt re of mas s ....................... . 247
10.4 Conservati on of li nea r moment um .................... . 250
10.5 Rocket moti on ............................... 251
10.6 Collision theory . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 255
10.7 Collision processes in th e zero-momentu m fram e . . . . . . . . . . . . . 259
10.8 The two-body proble m . . . . . . .. . . . . . . . . . . . . . . . . . . . 264
10.9 Two-body scattering . . . . . . . .. . . . . . . . . . . . . . . . . . . . 269
10.1 0 Integrable mechanical system s . ...................... 273
Appendix A Modelling bodies by particle s . . . . . . . . . . . . . . . . . . . 277
Problem s .................................. 279
11 The angula r momentum principle 286
11.1 The moment of a force ........................... 286
11.2 Angula r momentum . . . . . . . .. . . . . . . . . . . . . . . . . . . . 289
11.3 Angula r momentum of a rigi d body . . . . . . . . . . . . . . . . . . . . 292
11.4 The angula r momentum principle .. . . . . . . . . . . . . . . . . . . . 294
11.5 Conser vatio n of angula r momentum . . . . . . . . . . . . . . . . . . . . 298
11.6 Planar rigi d body motion . . . . . .. . . . . . . . . . . . . . . . . . . . 306
11.7 Rigi d body statics in thre e dimensions . . . . . . . . . . . . . . . . . . . 313
Problem s .................................. 317
viii Contents
3 Analytical mechanics 321
12 Lagrange’s equation s and conservatio n principle s 323
12.1 Constraints an d constrain t forces ...................... 323
12.2 Generali sed coordinates .......................... 325
12.3 Configuration spac e ( q –space ) . . . . . . . . . . . . . . . . . . . .. . . 330
12.4 D’Alembert ’s principle ........................... 333
12.5 Lagrange’s equati ons ........................... . 335
12.6 System s with moving constraint s . ................... . 344
12.7 The Lagrangian ............................... 348
12.8 The energy functio n h ........................... 351
12.9 Generali sed momenta ........................... 354
12.1 0 Symmetr y an d conser vatio n principle s . .................. 356
Problem s .................................. 361
13 The calculus of variations and Hamilton’s principle 366
13.1 Som e typica l minimisatio n problems . . . . . . . . . . . . . . . .. . . 367
13.2 The Euler–Lagrange equati on ....................... 369
13.3 Variati onal principles ........................... . 380
13.4 Hamilton ’s principle . . . .. . . . . . . . . . . . . . . . . . . . .. . . 383
Problem s .................................. 388
14 Hamilton’s equation s and phase spac e 393
14.1 System s of first orde r ODEs . . . . . . . . . . . . . . . . . . . . .. . . 393
14.2 Legendre t rans forms ........................... . 396
14.3 Hamilton ’s equations . . . .. . . . . . . . . . . . . . . . . . . . .. . . 400
14.4 Hamiltonian phas e spac e (( q , p )–space) . . . . . . . . . . . . . . .. . . 406
14.5 Liouville ’s theore m an d recurrenc e . . . . . . . . . . . . . . . . .. . . 408
Problem s .................................. 413
4 Furthe r topics 419
15 The genera l theory of small oscillations 421
15.1 Stable equilibriu m an d smal l oscillations . . . . . . . . . . . . . .. . . 421
15.2 The approximat e form s of T an d V ................... . 425
15.3 The genera l theory of norma l modes . . . . . . . . . . . . . . . . .. . . 429
15.4 Exi stence theory for normal modes ................... . 433
15.5 Som e typica l norma l mode problems . . . . . . . . . . . . . . . .. . . 436
15.6 Orthogonality of norma l modes . . . . . . . . . . . . . . . . . . .. . . 444
15.7 Genera l smal l oscillations .. . . . . . . . . . . . . . . . . . . . .. . . 447
15.8 Normal coordinates ........................... . 448
Problem s .................................. 452
Contents ix
16 Vecto r angula r velocity and rigi d bod y kinematics 457
16.1 Rotatio n abou t a fixed axis . . . . .. . . . . . . . . . . . . . . . . . . . 457
16.2 Genera l rigi d body kinematic s . . .. . . . . . . . . . . . . . . . . . . . 460
Problem s .................................. 467
17 Rotating reference frame s 469
17.1 Trans formati on formulae .......................... 469
17.2 Particle dynamics in a non-inertial fram e . . . . . . . . . . . . . . . . . 476
17.3 Motion relat ive to th e Eart h . ...................... . 478
17.4 Multi-particle system in a non-inertial fram e . . . . . . . . . . . . . . . 485
Problem s .................................. 489
18 Tenso r algebr a and th e inerti a tensor 492
18.1 Orthogonal transformation s . . . .. . . . . . . . . . . . . . . . . . . . 493
18.2 Rotated an d reflecte d coordinat e system s . . . . . . . . . . . . . . . . . 495
18.3 Scalars , vector s an d tensor s . ...................... . 499
18.4 Tenso r algebr a .............................. . 505
18.5 The inertia tensor .............................. 508
18.6 Principal axes of a symmetri c tensor ................... . 514
18.7 Dynamical symmetr y . .......................... . 516
Problem s .................................. 519
19 Problem s in rigi d bod y dynamic s 522
19.1 Equations of rigi d body dynamics .. . . . . . . . . . . . . . . . . . . . 522
19.2 Motion of ‘spheres’ ........................... . 524
19.3 The snoo ker ball . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 525
19.4 Free motion of bodies with axia l symmetr y . . . . . . . . . . . . . . . . 527
19.5 The sp inning top .............................. 531
19.6 Lagrangian dynamics of th e to p . .. . . . . . . . . . . . . . . . . . . . 535
19.7 The gyrocompass .............................. 541
19.8 Euler’s equati ons .............................. 544
19.9 Free motion of an unsymmetrica l body . . . . . . . . . . . . . . . . . . 549
19.1 0 The rolling wheel . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 556
Problem s .................................. 560
Appendi x Centres of mass and moments of inerti a 564
A.1 Cent re of mass ............................... 564
A.2 Moment of inerti a ............................ . 567
A.3 Parall el and perp endi cul ar axes ...................... 571
Answers to th e problem s 576
Bibli ography ................................... . 589
Index ....................................... 591
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